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Additivity properties for Value-at-Risk under Archimedean dependence and heavy-tailedness

Author

Listed:
  • Embrechts, Paul
  • Neslehová, Johanna
  • Wüthrich, Mario V.

Abstract

Mainly due to new capital adequacy standards for banking and insurance, an increased interest exists in the aggregation properties of risk measures like Value-at-Risk (VaR). We show how VaR can change from sub to superadditivity depending on the properties of the underlying model. Mainly, the switch from a finite to an infinite mean model gives a completely different asymptotic behaviour. Our main result proves a conjecture made in Barbe et al. [Barbe, P., Fougères, A.L., Genest, C., 2006. On the tail behavior of sums of dependent risks. ASTIN Bull. 36(2), 361-374].

Suggested Citation

  • Embrechts, Paul & Neslehová, Johanna & Wüthrich, Mario V., 2009. "Additivity properties for Value-at-Risk under Archimedean dependence and heavy-tailedness," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 164-169, April.
  • Handle: RePEc:eee:insuma:v:44:y:2009:i:2:p:164-169
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    References listed on IDEAS

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    1. Casper G. de Vries & Gennady Samorodnitsky & Bjørn N. Jorgensen & Sarma Mandira & Jon Danielsson, 2005. "Subadditivity Re–Examined: the Case for Value-at-Risk," FMG Discussion Papers dp549, Financial Markets Group.
    2. J. Dhaene & R. J. A. Laeven & S. Vanduffel & G. Darkiewicz & M. J. Goovaerts, 2008. "Can a Coherent Risk Measure Be Too Subadditive?," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 75(2), pages 365-386.
    3. repec:sae:ecolab:v:16:y:2006:i:2:p:1-2 is not listed on IDEAS
    4. Degen, Matthias & Embrechts, Paul & Lambrigger, Dominik D., 2007. "The Quantitative Modeling of Operational Risk: Between G-and-H and EVT," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 37(02), pages 265-291, November.
    5. Nikolay Nenovsky & S. Statev, 2006. "Introduction," Post-Print halshs-00260898, HAL.
    6. Wei, Gang & Hu, Taizhong, 2002. "Supermodular dependence ordering on a class of multivariate copulas," Statistics & Probability Letters, Elsevier, vol. 57(4), pages 375-385, May.
    7. Alink, Stan & Lowe, Matthias & V. Wuthrich, Mario, 2004. "Diversification of aggregate dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 35(1), pages 77-95, August.
    8. Marco Moscadelli, 2004. "The modelling of operational risk: experience with the analysis of the data collected by the Basel Committee," Temi di discussione (Economic working papers) 517, Bank of Italy, Economic Research and International Relations Area.
    9. Muller, Alfred, 1997. "Stop-loss order for portfolios of dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 21(3), pages 219-223, December.
    10. Barbe, Philippe & Fougères, Anne-Laure & Genest, Christian, 2006. "On the Tail Behavior of Sums of Dependent Risks," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 36(02), pages 361-373, November.
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    Citations

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    Cited by:

    1. Li, Haijun & Wu, Peiling, 2013. "Extremal dependence of copulas: A tail density approach," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 99-111.
    2. Chen Zou, 2009. "Dependence structure of risk factors and diversification effects," DNB Working Papers 219, Netherlands Central Bank, Research Department.
    3. Chuancun Yin & Dan Zhu, 2015. "New class of distortion risk measures and their tail asymptotics with emphasis on VaR," Papers 1503.08586, arXiv.org, revised Mar 2016.
    4. Mainik Georg & Rüschendorf Ludger, 2012. "Ordering of multivariate risk models with respect to extreme portfolio losses," Statistics & Risk Modeling, De Gruyter, vol. 29(1), pages 73-106, March.
    5. Suzanne Emmer & Marie Kratz & Dirk Tasche, 2013. "What Is the Best Risk Measure in Practice? A Comparison of Standard Measures," Working Papers hal-00921283, HAL.
    6. Georg Mainik & Ludger Rüschendorf, 2010. "On optimal portfolio diversification with respect to extreme risks," Finance and Stochastics, Springer, vol. 14(4), pages 593-623, December.
    7. Susanne Emmer & Marie Kratz & Dirk Tasche, 2013. "What is the best risk measure in practice? A comparison of standard measures," Papers 1312.1645, arXiv.org, revised Apr 2015.
    8. repec:hal:journl:hal-00921283 is not listed on IDEAS
    9. Tong, Bin & Wu, Chongfeng & Xu, Weidong, 2012. "Risk concentration of aggregated dependent risks: The second-order properties," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 139-149.
    10. Constantinescu, Corina & Hashorva, Enkelejd & Ji, Lanpeng, 2011. "Archimedean copulas in finite and infinite dimensions—with application to ruin problems," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 487-495.
    11. Umberto Cherubini & Paolo Neri, 2017. "Value-at-Risk Diversification of $\alpha$-stable Risks: The Tail-Dependence Puzzle," Papers 1704.07235, arXiv.org.
    12. Mao, Tiantian & Lv, Wenhua & Hu, Taizhong, 2012. "Second-order expansions of the risk concentration based on CTE," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 449-456.
    13. Ibragimov, Rustam & Prokhorov, Artem, 2016. "Heavy tails and copulas: Limits of diversification revisited," Economics Letters, Elsevier, vol. 149(C), pages 102-107.
    14. Cuberos A. & Masiello E. & Maume-Deschamps V., 2015. "High level quantile approximations of sums of risks," Dependence Modeling, De Gruyter Open, vol. 3(1), pages 1-18, October.
    15. Lv, Wenhua & Pan, Xiaoqing & Hu, Taizhong, 2013. "Asymptotics of the risk concentration based on the tail distortion risk measure," Statistics & Probability Letters, Elsevier, vol. 83(12), pages 2703-2710.
    16. Coqueret, Guillaume, 2014. "Second order risk aggregation with the Bernstein copula," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 150-158.
    17. Asimit, Alexandru V. & Gerrard, Russell, 2016. "On the worst and least possible asymptotic dependence," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 218-234.
    18. Chen, Die & Mao, Tiantian & Pan, Xiaoqing & Hu, Taizhong, 2012. "Extreme value behavior of aggregate dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 99-108.
    19. Guillén, Montserrat & Sarabia, José María & Prieto, Faustino, 2013. "Simple risk measure calculations for sums of positive random variables," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 273-280.
    20. Pavel V. Shevchenko, 2009. "Implementing Loss Distribution Approach for Operational Risk," Papers 0904.1805, arXiv.org, revised Jul 2009.

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